sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(916, base_ring=CyclotomicField(38))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,36]))
pari: [g,chi] = znchar(Mod(161,916))
Basic properties
| Modulus: | \(916\) | |
| Conductor: | \(229\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(19\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{229}(161,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 916.m
\(\chi_{916}(17,\cdot)\) \(\chi_{916}(53,\cdot)\) \(\chi_{916}(57,\cdot)\) \(\chi_{916}(61,\cdot)\) \(\chi_{916}(121,\cdot)\) \(\chi_{916}(161,\cdot)\) \(\chi_{916}(165,\cdot)\) \(\chi_{916}(225,\cdot)\) \(\chi_{916}(245,\cdot)\) \(\chi_{916}(273,\cdot)\) \(\chi_{916}(289,\cdot)\) \(\chi_{916}(333,\cdot)\) \(\chi_{916}(485,\cdot)\) \(\chi_{916}(501,\cdot)\) \(\chi_{916}(661,\cdot)\) \(\chi_{916}(729,\cdot)\) \(\chi_{916}(901,\cdot)\) \(\chi_{916}(905,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{19})\) |
| Fixed field: | 19.19.2999429662895796650415561622892044448017561.1 |
Values on generators
\((459,693)\) → \((1,e\left(\frac{18}{19}\right))\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \(1\) | \(1\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{916}(161,\cdot)) = \sum_{r\in \Z/916\Z} \chi_{916}(161,r) e\left(\frac{r}{458}\right) = 5.4003715158+29.7797915958i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{916}(161,\cdot),\chi_{916}(1,\cdot)) = \sum_{r\in \Z/916\Z} \chi_{916}(161,r) \chi_{916}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{916}(161,·))
= \sum_{r \in \Z/916\Z}
\chi_{916}(161,r) e\left(\frac{1 r + 2 r^{-1}}{916}\right)
= 0.0 \)